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Proof step 12 from proof euler-four-square-identity (previous step)

⊦ (i1 ^2 * j1 ^2 + i2 ^2 * j1 ^2 + (i3 ^2 + i4 ^2) * j1 ^2) + (i1 ^2 + i2 ^2 + i3 ^2 + i4 ^2) * (j2 ^2 + j3 ^2 + j4 ^2) = ((i1 ^2 * j1 ^2 - 2 * (i1 * j1) * ((i2 * j2 + i3 * j3) + i4 * j4)) + ((i2 * j2 + i3 * j3) + i4 * j4)^2) + ((i1 * j2 + i2 * j1) + i3 * j4 - i4 * j3)^2 + (i4 * j2 + i3 * j1 + i1 * j3 - i2 * j4)^2 + (i4 * j1 + i1 * j4 + i2 * j3 - i3 * j2)^2

User applied rule simplifier

And obtained the following new goals

Proof step 13:
⊦ (i4 * j1 + i1 * j4 + i2 * j3 - i3 * j2)^2 + (i4 * j2 + i3 * j1 + i1 * j3 - i2 * j4)^2 + ((i1 * j2 + i2 * j1) + i3 * j4 - i4 * j3)^2 + ((i2 * j2 + i3 * j3) + i4 * j4)^2 = i2 ^2 * j1 ^2 + (i3 ^2 + i4 ^2) * j1 ^2 + (i1 ^2 + i2 ^2 + i3 ^2 + i4 ^2) * (j2 ^2 + j3 ^2 + j4 ^2) + 2 * (i1 * j1) * ((i2 * j2 + i3 * j3) + i4 * j4)

Rule argument

Additional information

Used simplifier rules: ⊦ i + j = i + i0 ↔ j = i0; (s)% ⊦ i - j = i0 ↔ i = i0 + j; (s)%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2% ⊦ i + (j - i0) = i + j - i0; (s)